In the previous section, we establish the theoretical property of the bias corrected smoothed decorrelated score when the underlying conditional density \(f(x|y,)\)  is \(\ell \) th order smooth. However, this smoothness parameter \(\ell \)  is typically unknown in practice, leading to the following two complications. First, in Assumption REF , a kernel function \(K\)  of the same order is applied, which implicitly requires the knowledge on the smoothness parameter \(\ell \) . In practice, the choice of kernel functions is often determined by the user's preference rather than the theory. Since high order kernels may exacerbate the problem of variability, choosing low order kernels of 2 or 4 is often recommended (even if the density is more smooth); see [1]}. Second, the optimal bandwidth \(\delta \asymp n^{-1/(2\ell + 1)}\)  that balances the asymptotic bias and variance of the decorrelated score in Lemma REF  also depends on the unknown \(\ell \) . It is well known from the nonparametric literature that the choice of bandwidth is an extremely important problem of both theoretical and practical values [2]}, [3]}, [4]}, [5]}, [6]}. In this section, we focus on how to choose the bandwidth \(\delta \)  in a data-driven manner. In view of the above discussion on the kernels, we assume that a low order kernel \(K\)  is chosen (for simplicity, we still denote its order by \(\ell \) ) and meanwhile the underlying conditional density has a higher order smoothness parameter.
